The expository article about octonions by John (Baez) that appeared in the AMS Bulletin (copy here, a web-site here) is one of the best pieces of mathematical exposition that I have ever seen. The octonions can be thought of as a system of numbers generalizing the quaternions. As with the quaternions, multiplication does not commute, and things are even worse, it’s not associative either. So, probably best not to try and think of these as “numbers”, but they do give a very remarkable exotic algebraic structure, one that explains all sorts of other exotic structures occurring in different areas of mathematics. The article beautifully explains a lot of the intricate story of how octonions connect up surprising phenomena in algebra, geometry, group theory and topology.

If you’re a mathematical physics mystic like myself, you’re susceptible to the belief that anything this mathematically deep, showing up in seemingly unrelated places, must somehow have something to do with physics. The story of octonions is closely related to the story of Clifford algebras, which are definitely a crucial part of physics, but it seems to me we’re still a long ways from truly understanding the role in physics of Clifford algebras, much less the more esoteric octonions. One thing that is fairly well understood is that the sequence of division algebras explains some of the structure of low-dimensional spin groups in Minkowski signature, through the isomorphisms:
SL(2,R)=Spin(2,1)
SL(2,C)=Spin(3,1)
SL(2,H)=Spin(5,1)
The octonion story is supposed to be the next in line, involving Spin(9,1), but made much trickier by the fact that SL(2,O) doesn’t really exist, since the octonions are non-associative.

Back in 1982, a very nice paper by Kugo and Townsend, Supersymmetry and the Division Algebras, explained some of this, ending up with some comments on the relation of octonions to d=10 super Yang-Mills and d=11 super-gravity. Baez and Huerta in 2009 wrote the very clear Division Algebras and Supersymmetry I, which explains how the existence of supersymmetry relies on algebraic identities that follow from the existence of the division algebras. Kugo-Townsend don’t mention string theory at all, and Baez-Huerta refers to superstrings just in passing, only really discussing supersymmetric QFT. There’s also Division Algebras and Supersymmetry II by Baez and Huerta from last year, with intriguing speculation about Lie n-algebras and what these might have to do with relations between octonions and 10 and 11 dimensional supergravity. For a nice expository paper about this stuff, see their An Invitation to Higher Gauge Theory.

In contrast to the tenuous or highly-speculative connections to string theory that appear in these sources, the Scientific American article engages in the all-too-familiar hype pattern. The headline argument is that octonions are important and interesting because they’re “The Strangest Numbers in String Theory”, even though they play only a minor role in the subject. It wouldn’t surprise me at all if octonions someday do end up playing an important role in a unified theory, but the rather obscure connection to the calculation of the critical dimension of the superstring that seems to be the main point of the Scientific American article isn’t a very convincing argument for such a role.

Somehow I suspect that those string theorists who were upset by Scientific American’s decision to publish speculation by Garrett Lisi about E8 and wrote in to complain, won’t be similarly upset to find this highly speculative material about the octonions appearing in the magazine.

27 Responses to This Week’s Hype

I don’t think it is sporting to say that octonions have played only a minor role in string theory. Any area where maximal supersymmetry, large exceptional Lie-Groups (F_4 and E_6-8), triality, G_2, Exceptional Jordan algebra or a host of other constructs have cropped up is, in essence, an octonionic branch of mathematical or physical theory. Even Bott periodicity uses the octonions (in disguise) to cycle through the other three division algebras.

While hardly a supporter of the string triumphalist movement, I think that string theorists have been at their best in calling the attention of the mathematics and traditional physics communities to the possibility of integrating octonionic mathematics with core geometric and physical theory.

Octonionic mathematics used to be on the ‘down low’ but has now made the leap out of the closet and into mainstream research.

It’s kind of like with E8: I’ve got mixed feelings about all these exotic structures, feeling I can’t be sure what’s a beautiful, fundamental structure, and what’s a complex piece of mathematical junk that managed to just barely hold on to some interesting structure for kind of random reasons. The string theorists also did a great publicity job for E8, although they seem to have lost interest in it.

At least with Garrett’s SciAm article on E8, you didn’t have to put up with a sales job implying string theory was the reason to take an interest in the subject.

I really did love John’s expository article on Octonions. It lays out beautifully the attractive part of the mathematics and how it hangs together. If that could be connected to physics, I’d be interested, but the only connection advertised in the SciAm piece is the rather obscure one about getting the supersymmetry algebra to work out a certain way in a certain dimension. It’s not even clear to me that this is what explains why 10 is the critical dimension, which is the claim repeatedly hammered home in the article.

I never understood the big deal with octonions. Sure, it is the last division algebra, but if you relax your axioms a little the Cayley-Dickson construction gives an infinite tower of increasingly uninteresting algebras:
n=1: Reals.
n=2: Complex numbers.
n=4: Quaternions, not commutative.
n=8: Octonions, not associative.
n=16: Sedenions, not alternative but power associative.
n=32: 32-ions?
…
I can see why you need to give up commutativity – things can be done in different order – but why is the division algebra property important? There might be a reason that octonions have not made it into physics in 150 years.

The octonion story is supposed to be the next in line, involving Spin(9,1), but made much trickier by the fact that SL(2,O) doesn’t really exist, since the octonions are non-associative.

PSL(2,O) does exist: you just have to be careful about it. The octonions aren’t associative, but the subalgebra generated by any two octonions is. So, you just need to avoid recklessly multiplying lots of numbers when you don’t really need to.

You can define the octonionic projective line and the group PSL(2,O) acting on it. You can even go ahead and define the octonionic projective plane and PSL(3,O), which turns out to be a certain real form of E6. But then the show stops: to define PSL(4,O) we’d really need the associative law. It’s fun to see how one “hits the wall” here.

This is explained in my octonions paper, though I could make it all much clearer now.

but the only connection advertised in the SciAm piece is the rather obscure one about getting the supersymmetry algebra to work out a certain way in a certain dimension.

It’s not at all “obscure” — within the narrow confines of string theory at least — that classical superstring Lagrangians rely for their supersymmetry on a magical identity that holds only in spacetimes of dimensions 3, 4, 6 and 10. It’s explained in Green, Schwarz and Witten’s textbook, for example.

And it’s been known for quite some time that these special dimensions are 2 more than the dimensions of the reals, complexes, quaternions and octonions. And in fact this is no coincidence: the existence of these number systems is the simplest explanation of what’s going on here.

So, don’t try to make it sound like an obscure yawn-inducing technicality about “some supersymmetry algebra working out a certain way in a certain dimension”. It’s a shocking and bizarre fact, which hits you in the face as soon as you start trying to learn about superstrings. It’s a fact that I’d been curious about for years. So when John Huerta finally made it really clear, it seemed worth explaining — in detail in some math papers, and in a popularized way in Scientific American.

But of course, none of this has anything to do with whether superstring theory is right as a theory of physics. The article says quite clearly that superstring theory makes no testable predictions:

At this point we should emphasize that string theory and M-theory have as of yet made no experimentally testable predictions. They are beautiful dreams — but so far only dreams. The universe we live in does not look 10- or 11-dimensional, and we have not seen any symmetry between matter and force particles. David Gross, one of the world’s leading experts on string theory, currently puts the odds of seeing some evidence for supersymmetry at CERN’s Large Hadron Collider at 50 percent. Skeptics say they are much less. Only time will tell.

I thought you’d quote that part. Oh well.

By way, I just won a case of scotch from Dave Ring: I’d bet him that the LHC wouldn’t discover “strong evidence for supersymmetry” in its first year of operation.

It’s not even clear to me that this is what explains why 10 is the critical dimension, which is the claim repeatedly hammered home in the article.

I don’t think the octonions explain why 10 dimensions is the critical dimension – at least, not now. It would be cool if they did. But so far, all I know is that classical superstrings favor dimensions 3, 4, 6 and 10, thanks to the four normed division algebras. Other considerations pick out the 10-dimensional case, which happens to be the octonionic one.

It’s a bit odd to say we “repeatedly hammer home” something about the critical dimension – that was certainly not our intention, and we never even use the phrase “critical dimension”. But maybe I can guess how you’d get that impression. It’s probably worth comparing what John Huerta and I originally wrote, to what came out in the final version.

We originally wrote:

For strings, when the number of extra directions is 1, 2, 4, or 8, we get supersymmetry. Why? Because then its vibrations can be described using numbers in a division algebra. But the total number of dimensions of space and time is 2 more than the number of extra dimensions. So, we get supersymmetry when the total number of dimensions is 3, 4, 6, or 10. One of these dimensions is time; the rest are space.

Curiously, when we fully take quantum mechanics into account, it appears that only the 10-dimensional theory is consistent. This is the theory that uses octonions. So, if string theory is right, the octonions are not a useless curiosity: on the contrary, they play a fundamental role in understanding spacetime, matter, and the forces of nature!

Here it’s pretty clear, I hope, that quantum considerations pick out the 10-dimensional theory—not some special fact about octonions. The “curiously” makes it clear that we don’t understand the connection to the octonions.

The final version says:

At any moment in time a string is a one-dimensional thing, like a curve or line. But this string traces out a two-dimensional surface as time passes. This evolution changes the dimensions in which supersymmetry naturally arises, by adding two — one for the string and one for time. Instead of supersymmetry in dimension one, two, four or eight, we get supersymmetry in dimension three, four, six or 10.

Coincidentally string theorists have for years been saying that only 10-dimensional versions of the theory are self-consistent. The rest suffer from glitches called anomalies, where computing the same thing in two different ways gives different answers. In anything other than 10 dimensions, string theory breaks down. But 10-dimensional string theory is, as we have just seen, the version of the theory that uses octonions. So if string theory is right, the octonions are not a useless curiosity: on the contrary, they provide the deep reason why the universe must have 10 dimensions: in 10 dimensions, matter and force particles are embodied in the same type of numbers—the octonions.

This “provide the deep reason why the universe must have 10 dimensions” makes it sound as if some special fact about octonions explains why the universe has 10 dimensions. So yeah, that’s bad. But if you read the beginning of the paragraph you’ll see that’s not what’s going on: it’s quantum considerations that pick out the number 10. And the “coincidentally” makes it clear that we don’t understand the connection.

If you saw how much editing and counter-editing were involved in, perhaps you’ll forgive us for letting that phrase slip past. I also don’t like the title, but I couldn’t think of a better one. I had to admit that our original proposed title, “The Octonions”, would to most readers be about as appealing as “The Metaphraxis” or “The Sexadent”.

Of course, it would be cool if we could find a link between the calculation that singles out the number 10 and the fact that the normed division algebras give an identity that lets you write down supersymmetric string theory Lagrangians in dimensions 3, 4, 6 and 10. It’s hard to find real “coincidence” of this magnitude in math. So, I suspect it’s just a matter of time before someone finds a deeper link. I’ve made some nice progress on this but not enough to talk about.

By the way, I have permission to put the Sci Am article on my website about a month after it comes out, and I’ll also put up the various drafts, just for the amusement of people who wonder how this sort of editing process works.

The division algebra property is not important. What’s important are two things.

First, you get a normed division algebra in dimension n if and only if you can find an n-dimensional spinor representation of Spin(n). This starts the interplay between spinors and vectors, which provides the special features of Lorentzian geometry in dimension n+2: namely, dimensions 3, 4, 6 and 10.

Second, a normed division algebra is alternative. The alternative law gives the spinor identity that makes supersymmetry work for super-Yang-Mills theory and classical superstrings in dimensions 3, 4, 6 and 10. It also gives the Jacobi identity for the exceptional Lie algebras F4, E6, E7 and E8 — which contain the Lorentz Lie algebras for dimensions 3, 4, 6 and 10. It also gives a 3-cocycle on the superPoincare groups in these special dimensions, which let John Huerta build ‘categorified’ Lie supergroups relevant to describing superstring theory as a higher gauge theory.

So, a lot of math interacts in a marvelous way when you have a normed division algebra.

None of this stuff works anymore when we move further up the Cayley-Dickson tower. I bet something interesting does work, and I’d love to know what it is, but I don’t know and nobody seems to be working on it.

Needless to say, I’m not talking about whether any of this stuff is useful for physics. I’m talking about math.

The split-octonions have already been used by Ferrara and Duff et al. in the description of black hole and string charge vectors in toroidally compactified M-theory (N=8 supergravity). See arXiv:1002.4223 [hep-th]

Thanks a lot for the detailed explanations. I should have made clear that this is a quite different case than the usual editions of “This Week’s Hype”, which just about always involve a claim that “we’ve found a way to test string theory”. Thankfully, there’s nothing at all like that here, and the material in the article about the relationship of string theory/supersymmetry to the real world was accurate.

I read the article and enjoyed it. From the SA headlines, it sounded hypey, but the article itself was not. I am guessing Baez and Huerta don’t control how SA chooses to headline the story, which of course will be done in a way that catches the most attention.

I have done some reading of octonions, and I read John’s paper on it. It is very good, but technical to me. From what I understand about octonions is that they are anti-associative for three oconions that are different (excluding a real scaling factor). Since, the ORDER of the octonions determines the value, has anyone tried to make a consistent theory of spacetime using octonions? The order would determine different possible outcomes, but nature would only choose one. Also since the order does determine output, this gives a before and after quality if we build a length of time by successive octonion multiplication.

Maybe you can try my blog article here… it’s a bit technical, but a lot less technical than our actual paper.

But here’s something really quick: the real numbers, complex numbers, quaternions and octonions don’t just let you write down the (classical) theories of supersymmetric strings in 3, 4, 6, and 10 – they also let you write down (classical) theories of supersymmetric 2-branes in dimensions 4, 6, 7 and 11! And the 11-dimensional octonionic case is believed to be relevant to “M-theory”, whatever that is.

Once you hear this, you should wonder about 3-branes in dimensions 5, 7, 8 and 12. But if you look at the old “brane scan” picture on that blog entry I’m pointing you to, you’ll see it doesn’t quite work as smoothly as that. In particular, John Huerta has done a bunch of calculations showing that the trick relating octonions to strings in 10 dimensions and 2-branes in 11 dimensions does not continue to work one dimension higher. Apparently the nonassociativity messes things up!

(It’s long been known by string theorists that something “ends” at dimension 11. However, we are focusing on a limited portion of the math, not the whole story physicists are interested in. So, the calculations John Huerta did may be new, or at least a little different than the usual story.)

By the way, John Huerta has gotten a postdoc position with Peter Bouwknegt at Australian National University in Canberra. That’s ‘close’ to where I’m working here in Singapore – meaning, only about a 12-hour flight. (In fact Australia isn’t close to anything.) So, I hope to continue doing a bit of work with him on octonionic puzzles. There are not many people with a good intuition for the octonions, and he’s one.

Okay, I know this is trite (and liable to get deleted) but I haven’t found the answer after a bit of survey. So how is “octonions” pronounced? Is it “Oc-‘tone-ions” or is it “‘oct-‘ung-ions”? (As in the In-and-Out: do you want a _lot_ of onions with that?)

“The string theorists also did a great publicity job for E8, although they seem to have lost interest in it.”

This has nothing to do with the E8 per se as a mathematical structure. They were forced to focus on Heterotic E8 since at that time it was the only road connecting String theory to phenomenology. With the advent of D-branes and flux compactifications though they were able to construct realistic string vacua in IIB/F-theory with all the moduli stabilized and even to connect these models to cosmology i.e. dS vacua (via KKLT), or warped D-Brane inflation (via KKLMMT) etc. On the other hand Heterotic has some know problems with respect to moduli stabilization and wasn’t able to compete with IIB/F-theory on all these fronts of phenomenology.

Of course this doesn’t really matter since as is well know all these theories are connected via dualities. There is only one String theory.

“If you’re a mathematical physics mystic like myself, you’re susceptible to the belief that anything this mathematically deep, showing up in seemingly unrelated places, must somehow have something to do with physics.”

This point of view is absolutely pervasive with particle/GUT theorists right now. There is an absolute disdain for non-rigorous mathematics in theory. The problem is that non-rigorous mathematics has often been the tool to advance theory in the first instance. Newton’s theories depended on a very non-rigorous theory of calculus. Advancements in quantum mechanics were made for decades before renormalization could be rigorously handled. Feynman’s initial methods for QED were an absolute mess of ad hoc rules and tricks. Only later were the rules formalized.

To make better theory, we don’t need better mathematics. We need better Physics. Sort out the math later.

This has nothing to do with rigorous vs. non-rigorous. The theories being studied by particle/GUT theorists are not rigorously well-defined, and there’s nothing wrong with that.

Many physicists believe that whatever mathematicians are working on, it’s irrelevant to them: they just need the right “physical idea”, with the only math needed maybe something like PDEs. Newton was not so foolish. He was at the cutting edge of developing both new mathematics and new physics, well aware that the kind of new ideas about mathematics being developed by Leibniz and others were going to be required to express the new ideas about physics that he was developing. If he had taken the attitude that “all these new-fangled ideas about math can’t be needed for physics”, surely the dynamics of particles can be expressed using basic algebra and the theory of functions (e.g. today’s high school pre-calculus math), he would not have gotten far.

The problem with what has come out of string theory is not that it is based on too much abstract mathematics, but that it is based on a physical idea that turned out to be wrong.

“The problem with what has come out of string theory is not that it is based on too much abstract mathematics, but that it is based on a physical idea that turned out to be wrong.”

No, that was the problem with string theory. The problem now is that the people doing string theory are so raptured with the beauty of the mathematics in string theory, that they think the theory must be correct. Which brings us back to your quote. People have stopped looking for other physical analogies because they think their rigorous math has to be correct.

String theory is not based on rigorous mathematics. No one has a consistent definition of the theory, even at a non-rigorous level. What people won’t give up on is not a piece of rigorous mathematics, but a physical idea: space-time has 10/11 dimensions and strings/M-theory branes unify particle physics and gravity.

Some interesting mathematics has come out of string theory (e.g. mirror symmetry), but this happened not by physicists coming up with rigorous mathematics, but by physicists coming up with non-rigorous, conjectural ideas about mathematics, which mathematicians then used as inspiration to come up with rigorous versions (of small parts of the original ideas).

Peter
Very good! I agree to your thought. I think you are right.
As you said, many physicists and mathematicians won’t give up string theory right now because that gives the inspiration to them, especially mathematicians.

I think you too don’t give up the string theory as mathematician
because that you and your people in mathematics can obtain the conjectural ideas from string theory.

Even if string theory is not good physics, it is good for mathematicians to long for getting conjectural ideas.

@Peter: To me it seems increasingly likely that string theory is the physics equivalent of what in markets we call a “speculative episode”. I think theoretical physics has suffered from badly exagerated expected returns on investment of time in string theory. And as usual those who are stuck in the investment till their necks will not be the first ones to blow the whistle that a market correction is coming. But I think for the rest of us … we might start to think about how we’re going to bail our colleagues out …